Optimal Solutions to Infinite-Player Stochastic Teams and Mean-Field Teams

TL;DR

This paper establishes conditions under which optimal policies for infinite-player stochastic teams can be derived as limits of finite-player solutions, extending to mean-field teams with symmetry and providing existence results.

Contribution

It introduces a framework connecting finite and infinite-player team optimality, relaxing conditions under symmetry, and proving existence of optimal policies for mean-field teams.

Findings

Optimal policies for infinite teams can be obtained as limits of finite-team policies.

Symmetry conditions allow relaxation of convergence requirements.

Theoretical results are supported by illustrative examples.

Abstract

We study stochastic static teams with countably infinite number of decision makers, with the goal of obtaining (globally) optimal policies under a decentralized information structure. We present sufficient conditions to connect the concepts of team optimality and person by person optimality for static teams with countably infinite number of decision makers. We show that under uniform integrability and uniform convergence conditions, an optimal policy for static teams with countably infinite number of decision makers can be established as the limit of sequences of optimal policies for static teams with $N$ decision makers as $N \to \infty$. Under the presence of a symmetry condition, we relax the conditions and this leads to optimality results for a large class of mean-field optimal team problems where the existing results have been limited to person-by-person-optimality and not global optimality (under strict decentralization). In particular, we establish the optimality of symmetric (i.e., identical) policies for such problems. As a further condition, this optimality result leads to an existence result for mean-field teams. We consider a number of illustrative examples where the theory is applied to setups with either infinitely many decision makers or an infinite-horizon stochastic control problem reduced to a static team.